411 REFERENCE · Wikibooks
Linear Algebra and the C Language/a03v
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/* ------------------------------------ */
/* Save as: c00a.c */
/* ------------------------------------ */
#include "v_a.h"
/* ------------------------------------ */
/* ------------------------------------ */
#define RA R3
#define CA C3
/* ------------------------------------ */
#define FACTOR_E +1.E-0
/* ------------------------------------ */
int main(void)
{
double tA[RA*CA]={
/* x**0 x**1 x**2 */
+1, +1, +1,
+4, +2, +1,
+9, +3, +1,
};
double tb[RA*C1]={
/* y */
6,
3,
5
};
double **A = ca_A_mR(tA, i_mR(RA,CA));
double **b = ca_A_mR(tb, i_mR(RA,C1));
double **Pinv = i_mR(CA,RA);
double **Pinvb = i_mR(CA,C1);
clrscrn();
printf(" Fitting a linear Curve to Data:\n\n");
printf(" A:");
p_mR(A, S5,P2,C7);
printf(" b:");
p_mR(b, S5,P2,C7);
printf(" Pinv = V invS_T U_T ");
Pinv_Rn_mR(A,Pinv,FACTOR_E);
pE_mR(Pinv, S12,P4,C10);
stop();
clrscrn();
printf(" Pinv b ");
mul_mR(Pinv,b,Pinvb);
p_mR(Pinvb, S10,P4,C10);
printf("\n The coefficients a, b, c of the curve are: \n\n"
" y = %+.2f*x^2 %+.2f*x %+.2f\n\n"
,Pinvb[R1][C1],Pinvb[R2][C1],Pinvb[R3][C1]);
stop();
f_mR(b);
f_mR(A);
f_mR(Pinv);
f_mR(Pinvb);
return 0;
}
/* ------------------------------------ */
/* ------------------------------------ */
Presentation :
Let's calculate the coefficients of a polynomial.
y = ax**2 + bx + c
Which passes through these three points.
x[1], y[1]
x[2], y[2]
x[3], y[3]
Using the points we obtain the matrix:
x**2 x**1 x**0 y
x[1]**2 x[1]**1 x[1]**0 y[1]
x[2]**2 x[2]**1 x[2]**0 y[2]
x[3]**2 x[3]**1 x[3]**0 y[3]
That we can write:
x**2 x 1 y
x[1]**2 x[1] 1 y[1]
x[2]**2 x[2] 1 y[2]
x[3]**2 x[3] 1 y[3]
Let's use the Pinv_Rn_mR() function to solve
the system that will give us the coefficients a, b, c
Screen output example:
Fitting a linear Curve to Data :
A :
+1.00 +1.00 +1.00
+4.00 +2.00 +1.00
+9.00 +3.00 +1.00
b :
+6.00
+3.00
+5.00
Pinv = V * invS_T * U_T
+5.0000e-01 -1.0000e+00 +5.0000e-01
-2.5000e+00 +4.0000e+00 -1.5000e+00
+3.0000e+00 -3.0000e+00 +1.0000e+00
Press return to continue.
Pinv * b
+2.5000
-10.5000
+14.0000
The coefficients a, b, c of the curve are :
y = +2.50*x^2 -10.50*x +14.00
Press return to continue.
Copy and paste in Octave:
function y = f (x)
y = +2.50*x^2 -10.50*x +14.00;
endfunction
f (+1)
f (+2)
f (+3)